# Chapter 5 Many-Electron Atoms 5-1 Total Angular Momentum Spin ```Chapter 5 Many-Electron Atoms
5-1 Total Angular Momentum
Spin magnetic quantum number: ms=&plusmn;
1
because electrons have two spin types.
2
Quantum number of spin angular momentum: s=
Spin angular momentum: S= s(s  1) =
1
2
3

2
Spin magnetic moment: μs=-eS/m
The z-component of spin angular momentum: Sz=ms  =&plusmn;
1

2
L= l (l  1) and S= s(s  1) are both quantum vectors. They can be coupled to
together: J=L+S.
The z-component of spin angular moment: μsz=&plusmn;
e
2m
Total angular momentum J: J= J ( J  1)
The z-component of total angular momentum: Jz= mJ 
For only one electron outside the inner shell: J=
Lz=ml  , Sz=ms   mj=ms&plusmn;ml, j=l&plusmn;s=l&plusmn;
j ( j  1) and Jz=mj  =Lz+Sz,
1
.
2
LS coupling: L=  Li , S=  S i , and J=L+S. The scheme holds for most atoms and
these atoms in a weak magnetic field. And the quantum numbers in LS coupling are
L= L( L  1) , Lz=mL  , S= S (S  1) , Sz=mS  , J= J ( J  1) , Jz=mJ 
Rules of LS coupling:
L=l1+l2, l1+l2-1, l1+l2-2, …, |l1-l2|
S=s1+s2, s1+s2-1, s1+s2-2, …, |s1-s2|
J=L+S, L+S-1, L+S-2, …, |L-S|
Eg. Find the possible values of the total angular-momentum quantum number J
under LS coupling of two atomic electrons whose orbital numbers are l1=1 and
l2=2.
(Sol.) L=1+2, 1+2-1, 1+2-2(=|2-1|)=3, 2, 1
1 1 1 1
1 1
S= + ,
+ -1(=| - |)=1, 0
2 2 2 2
2 2
J=3+1, 3+1-1=3+0=2+1, 3+1-2=2+0=1+1, …, |1-1|=4, 3, 2, 1, 0
Effect of magnetic filed:
jj coupling: Ji=Li+Si, J=  J i . The scheme holds for heavier atoms and these atoms
in a strong magnetic field.
Representation of electron states: n2s+1LJ, where L=0(S), 1(P), 2(D), 3(F), 4(G),
5(H), 6(I), …
Eg. Find the values of s, L, and J for the representation of electron states: 1S0, 3P2,
2D , 5F , 6H .
3/2
5
5/2
(Sol.) 1S0: 1=2s+1  s=0; S  L=0; 0  J=0
3=2s+1  s=1; P  L=1; 2  J=2
2D : 2=2s+1  s=1/2; D  L=2; 3/2  J=3/2
3/2
3P :
2
5=2s+1  s=2; F  L=3; 5  J=5
6H : 6=2s+1  s=5/2; H  L=5; 5/2  J=5/2
5/2
5F :
5
Selection rules of LS coupling: △L=0, &plusmn;1, △J=0, &plusmn;1 (but J: 0→0 is prohibited),
△S=0
5-2 Pauli’s Exclusion Principle and Periodical Table
Pauli’s Exclusion Principle: No two electrons in one atom (or in close atoms) can
occupy the same quantum state.
Consider that particle 1 is in quantum state a and particle 2 is in quantum state b.
The wavefunctions ΨI=Ψa(1)Ψb(2) and ΨII=Ψa(2)Ψb(1) are identical to each other if
particle 1 and particle 2 are indistinguishable. Let an antisymmetric function be
ΨA(1,2)=[Ψa(1)Ψb(2)-Ψa(2)Ψb(1)]/√2=-ΨA(2,1) and a symmetric function be
ΨS(1,2)=[Ψa(1)Ψb(2)+Ψa(2)Ψb(1)]/√2=ΨS(2,1). If a=b, then ΨA(1,2)=0. That is,
particles 1 and 2 can not exist in the same quantum state.
Antisymmetric wavefunction: ΨA(1,2)=[Ψa(1)Ψb(2)-Ψa(2)Ψb(1)]/√2. It is in
agreement with Pauli’s Exclusion Principle.
Symmetric wavefunction: ΨS(1,2)=[Ψa(1)Ψb(2)+Ψa(2)Ψb(1)]/√2
Fermions: Particles of odd half-integral spin have antisymmetric wavefunctions.
They obey Pauli’s Exclusion Principle.
Eg. Electrons, Protons, Neutrons, etc.
Bosons: Particles of 0 or integral spin have symmetric wavefunctions. They do not
obey Pauli’s Exclusion Principle.
Eg. Photons, He atom in very low temperature, α-particle, etc.
Magic numbers: There are 2, 10, 18, 36, 54, and 86 electrons in the electron shells
such that the electron shells are exceptionally stable.
Periodical Table:
Atomic shells: n=1(K), 2(L), 3(M), 4(N), 5(O), …
For a fixed n, l=0, 1, 2, …, n-1. ml=0, &plusmn;1, &plusmn;2, …,
&plusmn;l. ms=&plusmn;
1
. The maximum number of electrons in
2
n 1
a shell is 2  (2l  1) =2n2.
l 0
Eg. n=1 shell can contain 2 electrons; n=2 shell can hold 8 electrons; n=3 shell
can hold 18 electrons, etc.
Electron configuration: s(l=0), p(l=1), d(l=2), …
Eg. Electron configuration of sodium (Na) is 1s22s22p63s1. It means that subshell
1s (n=1, l=0) and subshell 2s (n=2, l=0) contain respective 2 electrons; subshell 2p
(n=2, l=1) contains 6 electrons; and subshell 3s (n=3, l=0) contains 1 electron.
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